Regression-based reference limits and their reliability: example on hemoglobin during the first year of life

¹ Central Laboratory, University Central Hospital of Turku, Kiinamyllynk 4–8, FIN-20520 Turku, Finland.

² Departments of Clinical Chemistry and
³ Statistics, University of Turku, FIN-20500 Turku, Finland.
^a Author for correspondence. Fax 358-2-2613920;

	Abstract

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Calculation of reference limits by regression analysis makes it unnecessary to partition the reference data into subgroups, and age-dependent limits can be estimated within as narrow age intervals as necessary. However, the reliability of the regression-based reference limits has not been considered before. To get valid regression-based confidence intervals (CIs) for reference limits, one must evaluate the convolution of two distributions. In this study, age-dependent reference limits with corresponding CIs were produced for blood hemoglobin concentrations over the age interval from newborns to 12 months. We describe how the variance associated with the reference limits can be estimated, and present a Table from which appropriate values can be chosen for the calculation of regression-based reference limits and exact CIs. Also, an equation for the calculation of approximate CIs is given. The data were modeled by linear regression in several cumulative age groups to find the transition zone where the slope changed. After defining this cutoff point, piecewise linear regression was applied. Reference limits and their CIs calculated by conventional and piecewise linear regression methods were almost the same in older age groups but differed significantly during the period of most rapid age-dependent changes, i.e., during the 2 months after birth.

	Introduction

Top Abstract Introduction Subjects and Methods statistical methods Results Discussion Appendix References

 
The concept of reference values and their determination is based on recommendations of the IFCC and the NCCLS. These recommendations are restricted to situations in which the limits are estimated as specified fractiles by parametric or nonparametric methods from an univariate distribution (1)(2). Many biochemical quantities are dependent on sex, age, and other known characteristics. The instruction that the IFCC and NCCLS documents give for such situations is to divide the reference values into homogeneous subgroups (1)(2). Though partitioning seems to be a simple principle, it includes certain problems. How does one define the appropriate cutoff points between the subgroups? Are the resulting subgroups large enough for the calculation of reliable reference limits?

Regression analysis has been used by several investigators for the estimation of age-dependent reference limits when the reference groups are so small that the resulting subsets are not large enough for the calculation of valid reference limits. For example, during early childhood the physiological changes occur over short time intervals, and it may be practically impossible to get a sufficient number of reference values to establish reference limits for each narrow age group.

Harris and Boyd (3) considered statistical criteria for partitioning data sets in a suitable manner to obtain separate reference limits. However, even after partitioning there may still exist dependency between analyte and, for example, age in some subgroups. In those situations the use of regression analysis has been suggested for the calculation of percentiles or prediction intervals to be considered as reference limits. However, the theoretical basis of reference intervals and prediction intervals is different. If reference limits are required, they must be estimated, i.e., parameters of the reference distribution are estimated. Conversely, prediction intervals are calculated for random variables, not parameters.

Irjala et al. (4) estimated reference limits for IgA, IgG, and IgM in serum in children ages 6 months to 14 years with first- or second-order regression models. If the variance of residuals was age-dependent, then they calculated age-specific standard deviations, which were used when age-related reference intervals were determined. If there was no age dependency in variance of residuals, conventional regression analysis was applied. In the study of Anderson et al. (5) a weighted regression analysis was applied for the determination of age-specific reference intervals for serum prostate-specific antigen (PSA) because the variability of serum PSA values increased with age.¹ Burritt et al. (6) defined age- and sex-specific reference intervals for 19 biologic variables in serum samples from children ages 1 to 22 years. They used polynomial regression for age-dependent analytes and the 2.5% and 97.5% percentiles constituted the reference interval. In the Table included in their publication they show reference intervals for several age subgroups. Estimated limits at the center of each age interval were considered as age-group-specific reference limits. They also used a slight modification in older children to have a smooth transition to adult concentrations.

Vicente et al. (7) established reference intervals for serum ferritin depending on age and sex. Data were analyzed within each gender separately and dependency on age was first examined by plotting ferritin values vs age. After visual inspection of data, subgrouping according to age was carried out, regression analysis performed as necessary, and 95% prediction intervals were used as reference intervals. Gallo et al. (8) presented a statistical method in which variables that affected reference intervals were considered. In this method variables such as sex, age, weight, and alcohol consumption with possible effect on serum urea were included in a regression model. The effects of age and sex and their interaction on cardiac enzymes was illustrated by regression analysis in the study of Kairisto et al. (9). The 95% age-specific prediction intervals were calculated and prediction limits were considered as reference limits. However, none of the above articles discussed the reliability of the derived age-specific reference limits.

Piecewise regression can be used after plotting the data, for example, on age to model it by linear regression. Gonchoroff et al. (10) used a piecewise linear regression model to estimate the reference interval for alkaline phosphatase in different age groups. Piecewise regression can also be applied in different cumulative age groups. From the different cumulative groups the best-fitting curves can be selected and the point(s) thereby found where the values begin to change to another direction.

Especially during the first months of life, many laboratory analytes—including hemoglobin (Hgb) and red blood cell indices—change rapidly. Fig. 1 shows this clearly for Hgb. It is obviously difficult to find any one regression model for the whole age interval. Nonlinear regression (i.e., nonlinear in the parameters) would be required. A simpler method would probably be more useful for examining the various age dependencies of different analytes. The visual inspection of plots to judge the fit of data to the derived limits is an important step.

View larger version (27K): [in this window] [in a new window]   Figure 1. Reference limits (2.5% and 97.5%) and 95% CIs of Hgb concentration (g/L) over the age interval 0–12 months.

In this paper we use piecewise regression analysis to calculate 95% age-specific reference intervals for Hgb in children ages 0–12 months. Hgb concentrations are known to decrease quickly during the first months of life, after which the concentrations start to increase again (11). We decided to use the first year of life to describe the use of linear piecewise regression in calculating age-specific reference limits. The conventional reference limits are point estimates from a reference sample group of the true limits, which could be calculated if data were available from the whole reference population. Hence the imprecision of the sample estimates must be considered by determining the confidence intervals (CIs). In this paper we determine the variance associated with reference limits and develop a Table from which the necessary values can be selected for the calculation of CIs of reference limits produced by the regression method. An equation from which approximate intervals can be calculated is also presented.

	Subjects and Methods

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subjects
Data were collected during a 2-year period from a total of 2347 hospitalized children ages 0–1 year. The reference subjects were retrospectively selected according to their diagnoses in the hospital discharge diagnoses database. The list of exclusion diagnoses was based on our previous study (12). After exclusion steps the remaining reference sample group consisted of 310 children ages 0–1 year (mean 5.34 months). The range of Hgb values in these reference subjects was 86–250 (mean value 128 g/L).

The study protocol was officially accepted at the University Hospital of Turku and was in accordance with the Helsinki Declaration of 1975, as revised in 1983.

analytical methods
Blood specimens were obtained either by skin puncture or venipuncture as part of routine examination of the children. The blood was collected into microtubes containing EDTA as an anticoagulant. Coulter Counter S-Series (S Plus VI and T-880, Coulter Electronic) or Technicon H6000 (Technicon Instruments Corp.) analyzers were used for the analysis. The Coulter Counter S Plus VI was used as a master instrument and other analyzers were calibrated against it to produce the equivalent results. For internal quality control, stable control specimens and retained patient specimens were used.

	statistical methods

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calculation of reference limits and cis
In regression analysis, independent variables (e.g., age) are used to predict a dependent variable (analyte in question). A regression equation with one independent variable is (13)

(1)

In a regression model, each observation (Y) is assumed to come from a normal distribution centered at the concentration (ß₀ß₁x_i) implied by the model with the same variance (). The regression-based upper reference limit for a specified age point x₀ is a quantile:

(2)

where Z is a fractile from standard normal distribution, usually ±1.96. Even though the mean square error (S) is an unbiased estimator of for the regression model, the square root S of S is not an unbiased estimator of . Because (n - p)S/ is distributed as with n - p degrees of freedom—where n is sample size and p is number of parameters—the following is obtained:

(3)

where—in this situation—a_n-p is approximately (14). The exact value can be calculated by using gamma function, but here the approximation is used. Thus the estimator for the Eq. 2 is:

(4)

where ₀ is the fitted value at age point x₀ and = a_n-pS is an unbiased estimator of . The variance of the estimator (Eq. 4 ) is:

(5)

The variance of the first part of the Eq. 5 is:

(6)

where x₀ is the age point at which reference limits are estimated and is the mean value of age and v = [1/n (x₀- ¯X)/_i(x_i- ¯X)]. The variance of in the latter part of Eq. 5 is:

(7)

In a simple linear regression situation with only one independent variable (p = 2, i.e., two regression parameters ß₀ and ß₁), the variance associated with the reference limit is

(8)

where Z is square of the fractile of the normal distribution. The square root of above equation is the standard error, which is needed when CIs are calculated. When appropriate values (q) from Table 1 are chosen, exact CIs for regression-based reference limits can be defined:

(9)

This result corresponds to the conventional reference distribution with _{0 1}x₀ as a substitute for the mean. The regression-based limits, likewise, correspond to the conventional reference limits, and intervals around upper and lower limits correspond to the conventional CIs for reference limits. The Appendix contains the formula from which appropriate values of q can be calculated.

An approximate CI that is based on the asymptotic normal distribution of ₀ is:

(10)

where Z' is a fractile from a normal distribution corresponding to the appropriate confidence level. In the Appendix an example and code of a SAS program to calculate reference limits and CIs in piecewise linear regression are given.

Because our main goal was to evaluate reference limits and their CIs, the percentile intervals calculated here are suitable. For other purposes tolerance intervals and prediction intervals may be used (15).

determination of the point of change
Because of the known change in age dependency of Hgb during the first months of life, six cumulated age groups were formed (0–1, 0–2, 0–3, 0–4, 0–5, and 0–6 months). The age interval, i.e., the used step, was determined by calculating the optimal window width according to formula (16):

W = optimal window width, A = min(s,R/1.34), s is the estimate of standard deviation, and n is the number of children. R is the interquartile range, i.e., the difference between third and first quartiles (16).

The point of change can be estimated, for example, by modeling the data in different cumulative age groups and by finding the best-fitting model, i.e., with greatest R (coefficient of determination) and minimal residual mean square. If the rest of the data fit well for linear regression, then the intersection of those regression lines is the point of change. If necessary, the extension to more than two piecewise regression lines is straightforward (13).

If the point where the slope of the regression model changes is known, one can determine an indicator variable that takes this change point into account and a piecewise regression model can be applied. The regression equation is:

(11)

where x_c is the change point and x_i2 is the indicator variable that has a value of 0 or 1 depending on the value of x_i1.

aptness of regression model
It is important that the statistical procedure includes a check for the regression. The regression models are based on the assumptions that the distribution of residuals is normal and their variance is constant (13). Hence it may be necessary to make a logarithmic (or other) transformation of the original measurements to fullfill these requirements. In this study the normality of residuals was checked by the Shapiro–Wilk statistic, and graphic analysis of residuals was done to provide information about constancy of residuals. Examination of residuals is important because residual plots allow deviations from linearity to be clearly seen. Formal testing for linearity can also be done, but graphic analysis of residuals is usually sufficient.

Diagnostic measures should be used for the detection of problematic observations, i.e., outliers and influential data points. Outliers and influential observations have no general definition, and their meaning varies from one author to another. In this study an observation was considered influential if it had a major influence on the fitted model, i.e., on the values of regression parameters or predicted values. We used Cook's influence statistic, high leverage points, dffits, and dfbetas statistics to detect influential observations (17)(18)(19). An observation was considered to be an outlier if its dependent variable was either much higher or lower than the dependent variable of other observations with similar independent variables. The studentized residuals were determined for that purpose (20).

All the statistical calculations were done by SAS^® for Windows 6.11 package (SAS Institute) and graphic presentation by Microcal Origin^TM for Windows (Microcal Software). Values for Table 1 were calculated by Mathematica^TM (Wolfram Research).

	Results

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Figure 1 shows that the minimum Hgb value is ~1–2 months, and after that the concentrations start to increase slowly. The greatest changes occur during first 6 months. Because the variability of Hgb values decreased with age, a logarithmic transformation on the Hgb values was applied. In each of the cumulative age groups a linear regression was fitted (see Table 2 ). The best-fitting models—with greatest R and minimal residual mean square—were examined by residual analysis. Residuals (observed value-fitted value) were plotted against fitted values and age.

The Shapiro–Wilk statistic was used to evaluate the normality of the distribution of residuals. Outliers and influential data points were identified. A linear regression with the age group from newborns to 2 months was found to fulfill all the above criteria best. A second regression model was fitted to the rest of the data and the aptness of this model was also investigated by the methods described above. The intersection of these two lines occurred at 1.6 months, and this was selected to be the point where the slope of the piecewise linear model changed.

Finally, the whole age interval was modeled by a linear piecewise regression method with a change point of 1.6 months. Residuals, outliers, and influential data points also were estimated in this final model. Five outlying observations with studentized residuals greater than the critical value of the Table ( = 0.10) derived by Lund (20) were detected. However, these values were not removed, as we did not have any specific clinical information that would justify their removal. In Fig. 1 the 2.5% and 97.5% reference limits and their CIs for Hgb values were fitted on the whole age interval by the described methods. In Fig. 2 the age interval is extended only through the first 3 months to better display the higher variability in newborns. Table 3 shows the numeric reference limits with corresponding CIs for different ages and age groups. The exact limits and intervals for a specified age point could be supplied by computer on the laboratory report.

View larger version (19K): [in this window] [in a new window]   Figure 2. Reference limits (2.5% and 97.5%) and 95% CIs of Hgb concentration (g/L) over the age interval 0–3 months.

Conventional reference limits with CIs are shown in Table 4 . These were calculated by the Refval program (from H. E. Solberg, Department of Clinical Chemistry, Rikshospitalet, Oslo, Norway) after division of data into two age groups (0–1.6 months and 1.6 months–1 year). Reference limits as reviewed by Tietz (21) and limits calculated by our method are also shown in that Table .

	Discussion

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Although regression is frequently used for the estimation of age-specific reference limits, we were not able to find any previous publications where exact CIs for reference limits had been evaluated. Harris and Boyd (22) introduced nonparametric and parametric methods that have been used in estimating and smoothing percentile curves as a function of age. Elveback and Taylor (23) presented some statistical methodology of estimating percentiles and CIs, and they give an equation that corresponds to our approximative CI, i.e., Eq. 10 . Royston (24) presented intervals that were approximate and that could be used only if the CI for the mean value of x in the sample were needed. Wright and Royston (25) presented a brief discussion of age-related reference intervals and only mentioned confidence bands that could be calculated from ± 2 (SE). We believe that evaluation of the reliability of regression-based reference limits is just as important as the evaluation of the reliability of conventional reference limits.

Determination of the minimum sample size when using regression is not straightforward because there are no power functions available for choosing sample size when reference limits are estimated by this approach. In this case the sample size, as well as the estimated variance, affect the width of the reference interval and CIs. A prerequisite for reference interval estimation is that the reference interval should remain stable, i.e., the width of reference interval should not be dependent on sample size. The method of linear models does not need a large sample size. Some suggestions about its application for reference interval estimation are presented by Royston (24). CIs become narrower as sample size increases. How precise CIs need to be (and, thus, the optimal sample size) must be estimated by the user. In reference interval estimation, the IFCC proposed that at least 120 observations are needed for reliable estimates (26). If nonparametric CIs are produced, this is indeed the minimum sample size.

In Fig. 3 , approximate and exact values have been fitted at different degrees of freedom with the function of v. Here, as in Table 1 , v has been chosen so that unreliable calculations are prevented, i.e., the maximum value of v is 0.5. One can see that approximate values are near the exact values when the sample size is as small as 20. The difference between the values ranges from 0.06 to 0.08. Thus, if CIs for regression-based reference limits are estimated, approximate values can be used even for relatively small sample sizes. The width of the CIs naturally depends on sample size. In our Hgb data the sample size is 310 and the estimated 97.5% reference limit with corresponding 95% CI at 2 months of age is 136 (132–140). If we assume that we have 40 degrees of freedom (i.e., sample size is 43), the limit and intervals would be 136 (129–144). At this sample size (43) the exact and approximate CIs are precisely the same. In our Hgb data when the sample size is decreased from 310 to 43, the width of the CI would increase about 7 g/L.

View larger version (16K): [in this window] [in a new window]   Figure 3. Exact (solid lines) and approximate (dotted lines) values needed when 95% CIs are calculated.

Thus it seems that a relatively small sample size is sufficient for the method. This is of great importance, e.g., when considering pediatric reference samples because of the difficulties in the collection of large reference sample groups. A distinct advantage in the use of piecewise regression analysis is that partitioning of the data into several subgroups becomes unnecessary and the sudden changes in reference limits at certain age limits can be avoided.

The estimate of the mean of Y is less precise when age is located farther away from its mean value. Thus CIs around the reference limits are wider the farther age is from its mean (13). However, this effect is only marginal when the estimation of limits and intervals is restricted to the same age interval from which the data originated. Because logarithmic transformation was applied, the upper and lower CIs are not equal. This occurs also in conventional reference interval determination upon retransformation of CIs to original scale.

Table 4 shows limits and intervals calculated by our method and by the Refval program. The derived limits by both methods are close to each other, especially when comparing children approaching the age of 1 year. The greatest differences in reference limits by regression and conventional methods can be seen at ages at which Hgb values show steep changes, i.e., in age groups 0–48 days. Further subgrouping with the conventional method is not possible because too few reference subjects would remain in each age group. Without considering the age effect, the lower reference limit with CI would be 98 (87–108) and upper limit 224 (213–234).

There are several ways to use regression analysis for the calculation of age-specific reference limits. Nonlinear regression models may be flexible but impractical to use especially over age periods such as newborns, puberty, or menopause, where some underlying physiological change may affect the age dependency. Separate regression models could be applied to specified age intervals, but this would introduce the problem of sudden changes in limits between different age groups (5)(27). The method used in this study prevents such difficulties. We still strongly recommend visual inspection of data in addition to statistical calculations and measures to check the appropriateness of the model.

The proposed method is practical and could be used for a variety of laboratory analytes that show dependency on age or other known characteristics. We recommend it be used in clinical laboratories to improve the quality of age-dependent reference limits. The necessary calculations and display of data can be done with microcomputers and basic statistical software.

	Appendix

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exact cis for reference limits
The exact CIs (Eq. 9 ) satisfy the following equation:

Inserting the expressions for Q₀ and ₀ and making rearragements gives

from which the following formula can be derived:

where

Some values for this distribution are given in Table 1 .

example
In our Hgb data after a logarithmic transformation was applied, S=0.0100. If 97.5% reference limit and 95% CIs are evaluated at 2 months of age, the following calculations are needed:

₀ = _{0 1}x₀ - ₂x, where ₂ = 0 since the indicator variable has a value of zero.

= (4.713 0.007x₀)

= 4.717, which is the mean value of Hgb at given age point

= ₀ ß₁x₀ Za_n-pS

= 4.913, which is the upper 95% reference limit at age 2 months.

CI_u = (_{0 1}x₀ Za_n-pS) Z'a_n-p[v Z(a_n-p - 1)]S

= 4.940, which is the upper 95% CI for the upper reference limit. Value of v is 0.0135.

CI₁ = (_{0 1}x₀ Za_n-pS) - Z'a_n-p[v Z(a_n-p - 1)]S

= 4.885, which is the lower 95% CI for the upper reference limit.

Because our sample size is >300, an approximate value can be used, and it is 1.96(1.0008)(0.1406) = 0.2758. Hence, after retransformation the 97.5% reference limit at 2 months of age with 95% CI is 136 (132–140).

sas program
This program calculates 2.5% and 97.5% reference limits and 95% CIs in piecewise linear regression.

options linesize=75 pagesize=56 nodate;

data limits;

n-310; /sample size/

nn=(1/n);

df=n-3; /degrees of freedom/

x1m=5.3384357048; /mean value of age in the sample/

x2m=-0.260042289; /mean value of indicator variable/

x11=0.0004267694; /value of x11 in inverse of X'X/

x12=-0.002225561; /value of x12 in inverse of X'X/

x21=-0.002225561; /value of x21 in inverse of X'X/

x22=0.024588304; /value of x22 in inverse of X'X/

s=0.10013; /root mean square error/

z=1.96; /fractile needed to calculate 95%/

z2=z2; /reference and confidence intervals/

b0=4.703196; /intercept term/

b1=0.007488; /point estimate ofß₁ (age)/

b2=-0.347497; /point estimate of ß₂ (indicator variable)/

do age=0.00 to 12.000 by 0.001; /age interval/

if age gt 1.571 then x1=0; /point of change in/

else x1=1; /our data is 1.571/

iv=(age-1.571)x1; /indicator variable/

diff1=age-x1m;

diff2=iv-x2m;

m1=(diff1x11)(diff2x21);

m2=(diff1x12)(diff2x22);

part1=m1diff1;

part2=m2diff2;

all=part1part2;

v=nnall

an=sqrt(df/(df-0.5));

an2=anan;

vza=v(z2(an2-1));

svza=sqrt(vza);

zas=zans;

pred=exp(b0(b1age)(b2iv));

Q2=(b0(b1age)(b2iv)(zas)); /upper ref. limit ln/

Q2a=exp(Q2); /upper ref. limit in original unit/

Q1=(b0(b1age)(b2iv)-(zas)); /lower ref. limit ln/

Q1a=exp(Q1); /lower ref. limit in original unit/

former=zassvza;

cilrl=exp(Q1-former); /lower confidence interval and/

cilru=exp(Q1former); /upper for lower reference limit/

ciurl=expQ2-former); /lower confidence interval and/

ciuru=exp(Q2former); /upper for upper reference limit/

output;

end;

run;

	Footnotes

 
¹ Nonstandard abbreviations: PSA, prostate-specific antigen; Hgb, hemoglobin; and CI, confidence interval.

	References

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